Why is the hair is only on one side?
This delves into the concept of normals, and mathematical planes. My short answer is to extrude the plane a bit, then recalculate all the normals in the Mesh menu (make sure to select everything first). This will turn the plane from a two dimensional area into a three-dimensional object, and will allow it to emit hair from both sides.
If you don’t want the plane to be solid at all, I recommend changing how the hair is emitted to not use normals.